Henry A. Spencer, son of Dr W.H. Spencer (MD), studied at Bristol Medical School and University College, and qualified as a member of the Royal College of Surgeons (MRCS) of England and a licentiate of the Royal College of Physicians (LRCP) of London in 1887-1888. He became a member of the British Medical Association and served for some time as house surgeon at Bath United Hospital and as surgeon with the Union Steam Ship Company. In due course he became senior physician at the Bristol Royal Infirmary and was elected a Fellow of the Royal Institute of Public Health. His hobbies included all sports. In May 1906 he married Alice Mabel Haviland.
Spencer came to the Cape Colony early in 1892, was licensed to practise in the colony on 12 March that year, and settled in Cape Town until 1894. During that time he published three medical papers: "Cases of iodism" (1892) and "Cases of Herpes Zoster" (1894) in the (second) South African Medical Journal, and "Bell's Paralysis associated with Herpes Zoster" (1894) in The Lancet. From 1896 until the outbreak of the Anglo-boer War (1899-1902) he practised in King William's Town and became a member of the (second) South African Medical Association. During the Anglo-Boer War he served as a civil surgeon with the Highland Brigade and was awarded the Queen's Medal with five clasps. In December 1900 he settled at Middelburg, Transvaal Colony, where he was licensed to practise by the new administration in 1902 and where he remained for the rest of his career. In 1901 he was appointed district surgeon at Middelburg and by 1908 was also medical officer of health and railway medical officer there. Three of his papers were published in the Transvaal Medical Journal during these years: "Death by lightning stroke" (1907/8, Vol. 3, pp. 118-123), "Gun-shot injury to the head; symptoms after six years; trephined" (1908/9, Vol. 4, pp. 65-66), and "Epidemic malaria" (1909/10, Vol. 5, pp. 134-142). In the last of these he used meteorological observations made at Middelburg to explain malaria epidemics in the region, as these were associated with wet and warm conditions. During 1905-1909 he was a meteorological observer (of rainfall, atmospheric pressure, air temperature and humidity) for the Meteorological Department of the Transvaal Colony. In 1914-1915, during World War I (1914-1918), he participated in the South West Africa Campaign as officer commanding of the hospitals at Kuruman and Keetmanshoop. He reached the rank of major in the Defence Force of the Union of South Africa.
Spencer was a life member of the South African Association for the Advancement of Science. At the association's annual congress held in Cape Town in 1910 he presented two short archaeological papers: "Bushman rock pictures at D'sjate" (near Lydenburg, Report, 1910, pp. 132-135) and "Ancient copper mine near D'sjate" (Ibid, pp. 358-359).